Emergent patterns in diffusive Turing-like systems with fractional-order operator

Abstract

Patterns obtained in abiotically homogeneous habitats are of specific interest due to the fact that they require an explanation based on the individual behavior of chemical or biological species. They are often referred to as ‘emergent patterns,’ which arise due to nonlinear interactions of species in spatial scales that are much more larger than the individuals characteristic scale. In this work, we examine the spatial pattern formation of diffusive fractional predator–prey models with different functional response. In the first model, we investigate the dynamics of the Riesz fractional predation of Holling type-II functional response with the prey Allee effects, while the second model describes prey-dependent functional response of Ivlev-case and fractional reaction–diffusion. In order to give good guidelines on the correct choice of parameters for numerical simulation experiment of full fractional-order reaction–diffusion systems, we discuss the dynamics of each system in the biologically meaningful region $u\ge 0$ and $v\ge 0$ and give conditions for the existence of Hopf bifurcation, and Turing instability with either homogeneous (zero-flux) boundary conditions which imply no external input or Dirichlet boundary conditions. A novel alternating direction implicit based on backward Euler scheme with either the homogeneous Neumann (zero-flux) or Dirichlet boundary is applied for the numerical solution. The performance of this method is compared with that of the shifted Grünwald formula in terms of accuracy and computational time. Numerical experiments which justify our theoretical findings exhibits some fractional-order controlled patterns of stripes, spots and chaotic spiral-like structures that are mostly found in animal coats.